Alejandro Adem, Yongbin Ruan, Bin Zhang
In this paper we define an associative stringy product for the twisted orbifold K-theory of a compact, almost complex orbifold X. This product is defined on the twisted K-theory of the inertia orbifold of X, where the twisting gerbe is assumed to be in the image of the inverse transgression map.
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In this paper, we define a stringy product on $K^*_{orb}(\XX) \otimes \C $, the orbifold K-theory of any almost complex presentable orbifold $\XX$. We establish that under this stringy product, the de-locaized Chern character ch_{deloc} : K^*_{orb}(\XX) \otimes \C \longrightarrow H^*_{CR}(\XX), after a canonical modification, is a ring isomorphism. Here $ H^*_{CR}(\XX)$ is the Chen-Ruan cohomology of $\XX$. The proof relies on an intrinsic description of the obstruction bundles in the construction of Chen-Ruan product. As an application, we investigate this stringy product on the equivariant K…
Recently, there is an explosive growth of activities to understand stringy properties of orbifolds. In this article, we survey some of recent developments.
In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we conside…
In this paper we show a Kunneth formula for Bredon cohomology for actions of a pullback of groups. We show how this formula can be used to compute orbifold twisted K-theory for some discrete twistings. Using that result we compute orbifold K-theory for some 6-dimensional orbifolds introduced by Vafa and Witten. These examples also show the limitations of the method.
We use equivariant methods to establish basic properties of orbifold K-theory. We introduce the notion of twisted orbifold K-theory in the presence of discrete torsion, and show how it can be explicitly computed for global quotients.
In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was the prediction that a certain physicist's Euler number of the orbifold must be equal to the Euler number of any of its crepant resolutions. This was soon related to the so called McKay correspondence in mathematics (cf. [McK]). Later developments include stringy Hodge numbers (cf. [Z], [BD]), mirror symmetry of Calabi-Yau orbifolds (cf. [Ro]), and most recently the Gromov-Witten invariants of symplectic orbifolds (cf. [CR1-2]). One common …
We study the loop spaces of the symmetric powers of an orbifold and use our results to define equivariant power operations in Tate K-theory. We prove that these power operations are elliptic and that the Witten genus is an H_oo map. As a corollary, we recover a formula by Dijkgraaf, Moore, Verlinde and Verlinde for the orbifold Witten genus of these symmetric powers. We outline some of the relationship between our power operations and notions from (generalized) Moonshine.
Starting with a $\mathbb{C}^*$-valued cocycle on the global quotient orbifold $X // G$, we apply transgression techniques for 2-gerbes, as developed by Lupercio and Uribe, to construct a gerbe on the orbifold loop space $\mathcal{L}(X//G)$. This gives a loop based definition of twisted equivariant Tate K-theory for finite groups that conforms to the definition that Luecke provides for compact, connected Lie groups. We relate our construction to Ganter's work on Moonshine and Huan's quasi-elliptic cohomology.
For a finitely generated discrete group $Γ$, the $Γ$-sectors of an orbifold $Q$ are a disjoint union of orbifolds corresponding to homomorphisms from $Γ$ into a groupoid presenting $Q$. Here, we show that the inertia orbifold and $k$-multi-sectors are special cases of the $Γ$-sectors, and that the $Γ$-sectors are orbifold covers of Leida's fixed-point sectors. In the case of a global quotient, we show that the $Γ$-sectors correspond to orbifolds considered by other authors for global quotient orbifolds as well as their direct generalization to the case of an orbifold given by a quotient by a L…
Twisted factorial Grothendieck polynomials and equivariant $K$-theory of weighted Grassmann orbifolds
本文研究加权格拉斯曼 orbifold 的等效K-理论中的Schubert类，引入扭曲阶乘Grothendieck多项式并证明其代表这些类，同时给出结构常数和限制公式。
We study the relationship between the twisted Orbifold K-theories ${^α}K_{orb}(\textsl{X})$ and ${^{α'}}K_{orb}(\textsl{Y})$ for two different twists $α\in Z^3(G;S^1)$ and $α'\in Z^3(G';S^1)$ of the Orbifolds $\textsl{X}=[*/G]$ and $\textsl{Y}=[*/G']$ respectively, for $G$ and $G'$ finite groups. We prove that under suitable hypothesis over the twisting $α'$ and the group $G'$ we obtain an isomorphism between these twisted K-theories.
The purpose of this paper is to introduce the notion of loop groupoid associated to a groupoid. After studying the general properties of the loop groupoid, we show how this notion provides a very natural geometric interpretation for the twisted sectors of an orbifold, and for the inner local systems introduced by Ruan by means of a natural generalization of the concept holonomy of a gerbe.
In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan's theory of twisted orbifold cohomology and the local system of Freed-Hopkins-Teleman in their work in twisted K-theory. In the case in which the orbifold is simply a manifold we recover previous results of Gawedzki and Brylinski.
In this paper we construct an explicit geometric model for the group of gerbes over an orbifold $X$. We show how from its curvature we can obtain its characteristic class in $H^3(X)$ via Chern-Weil theory. For an arbitrary gerbe $\LL$, a twisting $\Korb{\LL}(X)$ of the orbifold $K$-theory of $X$ is constructed, and shown to generalize previous twisting by Rosenberg, Witten, Atiyah-Segal and Bowknegt et. al. in the smooth case and by Adem-Ruan for discrete torsion on an orbifold.
In this paper we review the combinatorics of the twisted sectors of $\mathcal{M}_{g,n}$, and we exhibit a formula for the age of each of them in terms of the combinatorial data. Then we show that orbifold cohomology of $\mathcal{M}_{g,n}$ when $g \to \infty$ reduces to its ordinary cohomology. We do this by showing that the twisted sector with minimal age is always the hyperelliptic twisted sector with all markings in the Weierstrass points; the age of the latter moduli space is just half its codimension in $\mathcal{M}_{g,n}$.
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